# What are some examples of problems in R but not in PR?

Also are there any R complete problems? These would be the hardest decidable problems to my knowledge. Sorry if this isn't specific enough.

PR class is associated with primitive recursive functions but not every total recursive function is a primitive recursive function. The most famous example of such functions is the Ackermann function. If you need to specify a decision problem then you can use e.g. the language $$L_{A} =\{ \langle m,n,k\rangle \mid m,n,k \in \mathbb{N}, A(m,n) \le k \} \subset \{0,1\}^{*}$$, (A - Ackermann function and e.g. $$\langle m,n,k\rangle = 1^{m}0^{n}1^{k}$$ or use any other encoding scheme). In this way you can build other examples including partial recursive functions (assuming that $$\bot < 0$$) which are also not in PR. Class R is closed under $$\le_{m}$$ reduction but any non-trivial language in R is R-complete under this reduction so you need to use some stronger reduction to find a specific set of the hardest problems in R.